What are some implications of Gödel's theorems on AI research?
Definitely there are a lot of implications for AI, including:


Inference with first-order-logic is semi-decidable. This is a big disappointment for all the folks that wanted to use logic as a primary AI tool.
Basic equivalence of two first-order logic statements is undecidable, which has implications for knowledge-based systems and databases. For example, optimisation of database queries is an undecidable problem because of this.
Equivalence of two context-free grammars is undecidable, which is a problem for formal linguistic approach toward language processing
When doing planning in AI, just finding a feasible plan is undecidable for some planning languages that are needed in practice.
When doing automatic program generation - we are faced with a bunch of decidability results, since any reasonable programming language is as powerful as a Turing machine.
Finally, all non-trivial questions about an expressive computing paradigm, such as Perti nets or cellular automata, are undecidable.